IHJPAS. 53 (4)2022 
 

213 

 This work is licensed under a Creative Commons Attribution 4.0 International License. 

 

  

 

 

Some Properties of Connectedness in Grill Topological Spaces 

 

 
 

 

 

 

  

 

 

                                                                                                             

 

 

Abstract 

 We use the idea of Grill, this study generalized a new sort of linked space like –connected –

hyperconnected and investigated its features, as well as the relationship between it and previously 

described notation. It also developed new sorts of functions, such as hyperconnected space, and 

identifying their relationship, by offering numerous instance and attributes that belong to this set. 

This set will serve as a starting point for further research into the sets many future possibilities. 

Also, we use some of the theorems and observations previously studied and relate them to the grill 

and the Alpha group, and benefit from them in order to obtain new results in this research. We 

applied the concept of Connected to them and obtained results related to Connected. The sources 

related to the Connected and Alpha were considered as starting points and an important basis in 

this research.   

 Keywords: Grill,  ₢*𝛼-connected, ₢*𝛼-𝑑𝑖𝑠connected, ₢∗𝛼𝑜(Ꝡ), ₢∗ 𝛼 -open.  

 

1. Introduction 

   [1], [2] established a grill notion in a topological space, and the grill has shown to be an effective 

tool for learning a variety of topological concerns. A subset of a topological space (Ꝡ, 𝜏) which 

is a non-empty collection ₢ is referred to be a grill whenever (a) Ⱥ ∈₢ and Ⱥ ⊆ Ӎ implying Ɓ∈₢. 

(b) Ꝡ has a subset Ⱥ and Ӎ also Ⱥ ∪ Ɓ∈₢ lead to Ⱥ ∈₢ or Ɓ∈₢. A triple (Ꝡ, 𝜏, ₢) topological 

space with grills is one type of topological space. 

Mukherjee and Roy [3] created a distinctive topology with a grill and investigated topological 

notions. For every topological space (Ꝡ, 𝜏) point Ꝡ, Neighborhoods are open of Ꝡ embodied 

Doi: 10.30526/35.4.2878 

Article history: Received 12 June 2022, Accepted 21 August 2022, Published in October 2022. 

 

 

 

Ibn Al Haitham Journal for Pure and Applied Sciences 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

 

Saad S. Suliman 

 Department of Mathematics, College of 

Education for Pure Science, Ibn Al 

Haitham,University of Baghdad, Iraq. 

saadsadeq05@gmail.com              

Esmaeel R.B.  

Department of Mathematics, College 

of Education for Pure Science, Ibn Al 

Haitham,University of Baghdad, Iraq. 

Ranamumosa@yahoo.com 

 

https://creativecommons.org/licenses/by/4.0/
mailto:saadsadeq05@gmail.com
mailto:Ranamumosa@yahoo.com


IHJPAS. 53 (4)2022 
 

214 

by 𝜏(Ꝡ). A mapping Ѱ:ℙ(Ꝡ) → ℙ(Ꝡ) is referred to as ∮( Ⱥ) = { Ꝡ ∈ Ꝡ: Ⱥ ∩ Ữ ∈ ₢ apiece 

Ữ ∈ τ(Ꝡ)} for every Ⱥ ∈ ℙ(Ꝡ). A mapping Ѱ: ℙ(Ꝡ) →ℙ(Ꝡ) is referred to as Ѱ (Ⱥ) = Ⱥ ∪ ∮ 

(Ⱥ) for every Ⱥ ∈ ℙ(Ꝡ). The map Ѱ Kuratowski closure axioms are met:  

 

      (𝑎) Ѱ (∅) = ∅, 

     (𝑏)   𝑤ℎ𝑒𝑛 Ⱥ ⊆  Ɓ, 𝑠𝑜 Ѱ (Ⱥ)  ⊆  Ѱ (Ɓ), 

     (𝑐)   𝑤ℎ𝑒𝑛 Ⱥ ⊆  Ꝡ, 𝑠𝑜 Ѱ (Ѱ (Ⱥ)) =  Ѱ (Ⱥ), 

     (𝑑)   𝑤ℎ𝑒𝑛 Ⱥ, Ɓ ⊆  Ꝡ, 𝑠𝑜 Ѱ (Ⱥ ∪  Ɓ) =  Ѱ (Ⱥ)  ∪  Ѱ (Ɓ). 

Grill topological spaces come in a variety of shapes and size as a discrete topology and a 

complement finite topology. 

[4] [5] On a space (Ꝡ, 𝜏), this agrees to a grill ₢ a topology exists 𝜏₢ on Ꝡ that is given by no 

one else by 𝜏₢ = {Ữ ⊆Ꝡ: Ѱ (Ꝡ – Ữ) = Ꝡ – Ữ}, Consequently, Ⱥ ⊆ Ꝡ, Ѱ (Ⱥ) = Ⱥ ∪ ∮( Ⱥ). τ ⊆ 

𝜏₢ and Ѱ (Ⱥ) = 𝑐𝜄(Ⱥ). Using the following as a basis, we are able to locate 𝜏₢ on Ꝡ by 

supplying 𝜏₢ through using the following as a basis 𝛽 (𝜏₢, Ꝡ) = {Ƴ − Ⱥ; Ƴ ∈ 𝜏, Ⱥ ∉ ₢}. 

[6]. On a space (Ꝡ, 𝜏), there is a grill ₢, 𝜏 ⊆ 𝛽(₢, τ) ⊆ 𝜏₢, where 𝛽(₢, τ) basis for 𝜏₢. 

As an example, [7] to exist in a space (Ꝡ, 𝜏) ,  𝜏₢ = 𝜏 whenever ₢ = ℙ(Ꝡ) ∖ {Ø}  

implying 𝜏₢ = τ. The family of all 𝛼 -open set is showed by 𝜏𝑠. 

𝛼 -𝑜𝑝𝑒𝑛 𝑖𝑠 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 Ⱥ 𝑜𝑓 𝑎 𝑠𝑝𝑎𝑐𝑒 (Ꝡ, 𝜏), [8] 𝑖𝑓 Ⱥ ⊆ 𝜍𝑙(ᶩ𝑛𝑡(Ⱥ)), 

Let (Ꝡ, τ, ₢) be topological space. The subset Ⱥ in Ꝡ is known as  ₢ 𝛼 -open if Ⱥ ⊆ Ѱ(ᶩ𝑛𝑡(Ⱥ)), 

and every Ѱ 𝛼 -open is an 𝛼 -open. Many researchers have generalized using these [9] [10]. The 

symbol ᶩ𝑛𝑡(Ⱥ) is the interior set of Ⱥ and 𝜍𝑙(Ⱥ) denotes the closure of  Ⱥ.  

combinations 

The space (Ꝡ, ) is disconnected if and only if there exist two open disjoint nonempty sets Ⱥ and 

Ɓ such that Ⱥ⋃Ɓ = Ꝡ. i.e., Ꝡ is disconnected 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 Ꝡ =  ⋃Ɓ ;  Ⱥ, Ɓ ∈   , Ⱥ ⋂ Ɓ  =

  , Ⱥ, Ɓ .  

The sets Ⱥ and Ɓ form a separation of Ꝡ. The space (Ꝡ, ) is connected if and only if it is not 

disconnectaaed. Ꝡ is connected if and only if Ꝡ  Ⱥ⋃Ɓ ;  Ⱥ, Ɓ ∈   , Ⱥ ⋂Ɓ =   , Ⱥ, Ɓ . 

 

2. Grill 𝜶 -open sets 

  

Definition  2.1. 

  In [10], the set Ⱥ is referred  to as Grill α-open if  ᶊ ∈ 𝜏 ; ᶊ − Ⱥ ∉  ₢ and Ⱥ − ᶩ𝑛𝑡𝜍𝑙₢(ᶊ) ∉ ₢. As 

stated by ₢∗α − open, the complment of ₢∗ α − open is ₢∗α − closed. The set of all ₢∗α −

open is represented by ₢∗α𝑜(ꝡ), and  ₢∗α − closed is denoted by ₢∗α𝑐(ꝡ) . 

a 

Example 2.2. [10] 

 Let (Ꝡ,ꚍ, ₢) is a topological space of the grill, and let  Ꝡ = {ꝡ
1

, ꝡ
2

, ꝡ
3

}, 𝜏 =

{Ꝡ, Ø, {ꝡ
1

}, {ꝡ
1

, ꝡ
2

}} , ℱ = {Ꝡ, Ø, {ꝡ
3

}, {ꝡ
2

, ꝡ
3

}} , ₢ = {ᶊ ⊆ Ꝡ; ꝡ
2

∊ ᶊ} , Ø: P(Ꝡ) ⇾ P(Ꝡ), 

}} ꝡ
2

},{ ꝡ
2

,ꝡ
3

},{ ꝡ
1

, ꝡ
2

,{Ø , Ꝡ={ ₢τ ,Ø ∪ Ⱥ = )Ⱥ(  Ѱ ,Ø(Ⱥ) = {ꝡ ∈ Ꝡ; ∀ᶊ ∈ 𝜏𝑥  ; ᶊ ∩ Ⱥ ∈ ₢}

,ℱ₢ = {Ꝡ, Ø,{ ꝡ3},{ꝡ1},{ꝡ2, ꝡ3}},so ₢
∗α𝑜(ꝡ) =

 {Ꝡ, Ø, {ꝡ
3

}, {ꝡ
2

}, {ꝡ
1

}, {ꝡ
2

, ꝡ
3

}, {ꝡ
1

, ꝡ
2

}, {ꝡ
3

, ꝡ
1

}}. 

 

 



IHJPAS. 53 (4)2022 
 

215 

 

Theorem 2.3. [10] 

    The union of any family of ₢∗ − 𝛼 open set is a ₢∗ − 𝛼 open set. 

 

Remark 2.4. [10] 

    Both ideas are related ₢∗α −  open set  , α −  open set are independent. 

 

Remark 2.5. [10] 

Supra topology refers to the collection of all ₢∗α-open sets. 

 

Remark 2.6 [10] 

 i. The open set leads to ₢∗α- open set. 

ii. The closed set leads to ₢∗α- closed set. 

 

Theorem 2.7. [10] 

   The ₢ α-open leads to ₢∗α-open. 

 

Proposition 2.8. [10] 

   For any Grill topology (Ꝡ, τ, ₢), Ⱥ is a ₢ α-open set if and only if Ⱥ 𝑖𝑠 𝑎 ₢∗α-open set whenever 

₢ = ℙ(Ꝡ) ∖ {∅}. 

 

 

Definition 2.9. [10] 

    The function   ᶂ: (Ꝡ, τ, ₢)) → (Ꝡ, 𝜏′, ₢) is referred to as:  

1. ₢∗α-open function, shortly "₢∗α-o function" if ᶂ(ᶊ) ∈ ₢∗𝛼𝑜(𝑦) when ᶊ ∈ ₢∗𝛼𝑜(Ꝡ). 

2. ₢∗∗α- open function, shortly "₢∗∗α-o function " if ᶂ(ᶊ) ∈ ₢∗𝛼𝑜(𝑦) when ᶊ ∈ 𝜏 . 

3. ₢∗∗∗α- open function, shortly "₢∗∗∗α-o function " if ᶂ(ᶊ) ∈ 𝜏′ whenever ᶊ ∈ ₢∗𝛼𝑜(Ꝡ). 

 

Definition 2.10  

   The function   ᶂ ∶ (Ҳ, 𝜏, ₢) → (Ꝡ, 𝜏′, ₢) is called ;  

1. ₢∗𝛼 −continuose function, shortly "₢
*
α-"continuous function\" " if ᶂ

−1(ᶊ) ∈

₢∗𝛼𝑂(Ҳ) for all ᶊ ∈ 𝜏′.     

2. Strongly ₢∗𝛼 − 𝑐ontinuose function shortly "strongly ₢*α-continuous function"  if ᶂ
−1(ᶊ) ∈

𝜏, fore every ᶊ ∈ ₢∗𝛼𝑂(Ꝡ).     

3. ₢∗𝛼 −irresolute  function, shortly "₢
*
α-irresolute function" if ᶂ

−1(ᶊ) ∈ ₢∗𝛼𝑂(Ҳ), for every ᶊ ∈

₢∗𝛼𝑂(Ꝡ)     

 

3. Grill 𝜶 -open sets in grill connected space 

 

Definition 3.1:  

   The space (Ꝡ, τ, ₢) is a ₢∗𝛼𝑜- disconnected if and only if there exist two ₢∗𝛼-open disjoint 

nonempty sets Ⱥ and Ɓ such that Ⱥ⋃Ɓ = Ꝡ. i.e.,Ꝡ is a ₢∗𝛼𝑜- disconnected if and only if Ꝡ 

=Ⱥ⋃Ɓ ; Ⱥ, Ɓ ∈ ₢∗𝑠𝑜(Ꝡ) , Ⱥ ⋂ Ɓ =   .  

The s𝑒ts Ⱥ and Ɓ form a ₢∗𝑠𝑜-separation of Ꝡ. The space (Ꝡ, τ, ₢) is a ₢∗𝑠𝑜- connected if and 



IHJPAS. 53 (4)2022 
 

216 

only if it is not ₢∗𝛼𝑜- disconnected. Ꝡ is a ₢∗𝛼𝑜- connected if and only if Ꝡ  Ⱥ⋃Ɓ ; Ⱥ, Ɓ ∈ 

₢∗𝛼𝑜(Ꝡ), Ⱥ ⋂ Ɓ =   , Ⱥ, Ɓ . 

 

Example 3.2. 

    L𝑒t (Ꝡ, τ, ₢) topological 𝑠pace b𝑒 a 𝑔𝑟ill a𝑛d  Ꝡ = {Ꝡ
1

, Ꝡ
2

, Ꝡ
3
}, 𝜏 = 𝜏₢ =

{Ꝡ, Ø, {Ꝡ
2
}, {Ꝡ

3
}}, ₢ = ℙ(Ꝡ) ∖ {Ø} is a ₢∗𝛼𝑜- connected space. 

 

Remark 3.3 

   Every disconnected set is a ₢∗𝛼𝑜-disconnected. 

 

Proof.  

  Let Ꝡ is disconnected set, then the exist 𝒲 ≠ ∅, 𝒵 ≠ ∅ and 𝒲, 𝒵 ∈ τ ; 𝒲 ⋂ 𝒵 =

∅  𝑎𝑛𝑑 𝒲 ⋃ 𝒵 = Ꝡ since every open set in ₢∗𝛼-open set. Therefore, Ꝡ is ₢∗𝛼𝑜-disconnected. 

 

Remark 3.4. 

   The space (Ꝡ, τ, ₢) is a ₢∗𝛼𝑜- disconnected with any grill and ₢∗𝛼𝑜(Ꝡ) = ℙ(Ꝡ) if  Ꝡ 

contained more than one element since there exist Ⱥ  and Ⱥ𝑐  ∈ ₢∗𝛼𝑜(Ꝡ); Ⱥ ⋃Ɓ = Ꝡ, Ⱥ ⋂ Ɓ 

=  , Ⱥ, Ɓ . 

 

Remark 3.5. 

    The space (Ꝡ, τ, ₢) is a ₢∗𝛼𝑜- disconnected and ₢ = ∅  and 𝜏₢ = {Ꝡ, Ø}, so ₢
∗𝛼𝑜- connected 

space. 

 

Remark 3.6.  

   If  𝜏₢ = ℱ₢ and  𝜏₢ ≠ Ι𝑛𝑑𝑖𝑠𝑐𝑟𝑒𝑡 , when  ₢ = ℙ(Ꝡ) ∖ {Ø}. This means  that (Ꝡ, τ, ₢) is a 

₢∗𝛼𝑜- disconnected .  

 

Theorem 3.7.  

  The space (Ꝡ, τ, ₢) is a ₢∗𝛼𝑜-connected if and only Ꝡ cannot be written as a union of two 

non-empty disjoint closed sets. 

 

Proof. 

   Let Ꝡ be a ₢∗𝛼𝑜-connected if  Ꝡ = Ⱥ ⋃Ɓ, such that; Ⱥ and Ɓ ∈ ₢∗𝛼𝑐(Ꝡ) , Ⱥ ⋂ Ɓ  =

 , Ⱥ, Ɓ  , so. Ⱥ = Ɓ𝑐  𝑎𝑛𝑑 Ɓ = Ⱥ𝑐  , then Ꝡ = Ⱥ⋃Ɓ; Ⱥ and Ɓ ∈ ₢∗𝛼𝑜(Ꝡ), Ⱥ ⋂ Ɓ =

 , Ⱥ, Ɓ   , that is mean that Ꝡ  is a ₢∗𝛼𝑜- disconnected. That is contradiction. Therefore, Ꝡ 

cannot be written as a union of two non-empty disjoint closed set. Now, if  Ꝡ is a ₢∗𝛼𝑜- 

disconnected space ,so. Ꝡ= Ⱥ ⋃Ɓ; Ⱥ and Ɓ ∈ ₢∗𝛼𝑜(Ꝡ), Ⱥ ⋂ Ɓ =  , Ⱥ, Ɓ  , so Ⱥ =

Ɓ
𝑐  𝑎𝑛𝑑 Ɓ = Ⱥ𝑐 ,then Ⱥ and Ɓ ∈ ₢∗𝛼𝑐(𝑥), but that is a contradiction. Therefore, Ꝡ is a ₢∗𝛼𝑜-

connected space.   

 

Theorem 3.8. 

   The space (Ꝡ, τ, ₢) is a ₢∗𝛼𝑜-connected if and only if the only the subsets of the space Ꝡ 

which are ₢∗𝛼-open and ₢∗𝛼-closed are Ꝡ and Ø . 

 



IHJPAS. 53 (4)2022 
 

217 

 

Proof.  

  Let Ⱥ and Ⱥ𝑐  ∈ ₢∗𝛼𝑜(𝑥) and Ⱥ ≠ Ꝡ , Ⱥ ≠ Ø ,so Ꝡ = Ⱥ⋃Ⱥ𝑐 ; Ⱥ ⋂ Ⱥ𝑐  =  , Ⱥ, Ⱥ𝑐 ≠ . 

Then, Ꝡ is a ₢∗𝛼𝑜-disconnected, and that is a contradiction. So, where Ⱥ ⊆ Ꝡ , Ⱥ , Ⱥ𝑐 ∈

₢∗𝛼𝑜(𝑥), then Ⱥ = Ꝡ or Ⱥ = Ø, let Ꝡ be a ₢∗𝛼𝑜-disconnected space. This means that Ꝡ =

𝒲⋃𝒵 ; 𝒲, 𝒵 ∈ ₢∗𝛼𝑜(𝑥), 𝒲 ⋂ 𝒵 =   and  𝒲, 𝒵 ≠ Ø implies that 𝒲 = 𝒵 𝑐  and 𝒵 = 𝒲𝑐 . So 

𝒲, 𝒵 ∈ ₢∗𝛼𝑐(𝑥) (that is contradiction) Therefore, Ꝡ is a ₢∗𝛼𝑜-connected space. 

      

Remark 3.9.  

   If ᶂ ∶ (Ꝡ, τ, ₢) → (Ƴ, 𝜏′, ₢) is a ₢∗𝛼𝑜-irresolute and onto function and Ƴ is a ₢∗𝛼𝑜-connected 

space then Ꝡ is not necessary ₢∗𝛼𝑜-connected space. 

 

Example 3.10. 

    Let ᶂ ∶ (ℝ, D, ₢) → (ℝ, Ι, ₢) ; ḟ(Ꝡ) = Ꝡ for all Ꝡ ∈ ℝ and ₢ = ℙ(Ꝡ) ∖ {Ø} ,so ḟ is a ₢∗𝛼𝑜-

 𝑖𝑟𝑟𝑒𝑠𝑜𝑙𝑢𝑡𝑒 and onto function and (ℝ, Ι, ₢) is a ₢∗𝛼𝑜-connected space and (ℝ, D, ₢) is a ₢∗𝛼𝑜-

disconnected space.      

 

Theorem 3.11.  

   If  Ⱥ and Ɓ are a ₢∗𝛼𝑜-connected spaces of (Ꝡ, τ, ₢) and Ⱥ ⋂ Ɓ ≠ , then         

Ⱥ⋃Ɓ is a ₢∗𝛼𝑜-connected space. 

 

Proof. 

    Let (Ꝡ, τ, ₢) be a grill topological space and  Ⱥ, Ɓ ⊆ Ꝡ ; Ⱥ, Ɓ be a ₢∗𝛼𝑜-connected space. 

Now, If  Ⱥ⋃Ɓ is a ₢∗𝛼𝑜-disconnected space, so  Ⱥ⋃Ɓ = 𝒲⋃𝒵 ; 𝒲, 𝒵 ∈

₢∗𝛼𝑜(𝑥)(Ⱥ⋃Ɓ), 𝒲 ⋂ 𝒵 =   and  𝒲, 𝒵 ≠ Ø ,then Ⱥ ⊆ Ⱥ⋃Ɓ , Ⱥ ⊆ 𝒲⋃𝒵, Ⱥ ⊆ 𝒲 𝑜𝑟 Ⱥ ⊆ 𝒵. 

Similarly, Ɓ leads to either Ⱥ ⊆ 𝒲 and Ɓ ⊆ 𝒲, then Ⱥ⋃Ɓ ⊆ 𝒲, then 𝒵 = Ø contradiction.  

Or, Ⱥ ⊆ 𝒵 and Ɓ ⊆ 𝒵, then Ⱥ⋃Ɓ ⊆ 𝒵, then 𝒲 = Ø C! Or Ⱥ ⊆ 𝒲 and Ɓ ⊆ 𝒵, then Ⱥ ∩ Ɓ ⊆

𝒲 ∩ 𝒵, then Ⱥ ∩ Ɓ = Ø C! Or Ⱥ ⊆ 𝒵 and Ɓ ⊆ 𝒲, then Ⱥ ∩ Ɓ ⊆ 𝒲 ∩ 𝒵, then Ⱥ ∩ Ɓ =

Ø. This is a contradiction. So, Ⱥ⋃Ɓ is a ₢∗𝛼𝑜-connected space. 

 

Remark 3.12.  

  We can generalize Theorem 3.11 to a family of a ₢∗𝛼𝑜-connected sets as follows: let {Ⱥ∝}∝∈∧ 

be a family of a ₢∗𝛼𝑜-connected subsets of a space (Ꝡ, τ, ₢) and ⋂ Ⱥ∝ ≠ Ø∝∈∧ , then ⋃ Ⱥ∝∝∈∧  is 

a ₢∗𝛼𝑜-connected set.   

 

 

4. Grill 𝜶-open sets in grill connected space hyperconnected 

  

Definition 4.1.  

  In any grill topological space, (Ꝡ, τ, ₢) is said to be: 

  1. ∗-hyperconnected if Ⱥ is τ₢-dense (𝜍𝑙₢(Ⱥ) = Ꝡ) for every non-empty open subset Ⱥ of Ꝡ. 

  2. ₢∗-hyperconnected if Ꝡ − 𝜍𝑙₢(Ⱥ) ∉ ₢ for every non-empty open subset Ⱥ of Ꝡ. 

  3. ₢∗𝛼-hyperconnected if Ꝡ − 𝜍𝑙₢(Ⱥ) ∉ ₢ for every non-empty ₢
∗𝛼-𝑜𝑝𝑒𝑛 subset Ⱥ of Ꝡ. 

  4. ₢∗𝛼𝑜-hyperconnected if 𝜍𝑙₢(Ⱥ) = Ꝡ for all Ⱥ ∈ ₢
∗𝛼𝑜(Ꝡ).  

 



IHJPAS. 53 (4)2022 
 

218 

Proposition 4.2.  

  1.Every ₢∗𝛼𝑜-hyperconnected is ∗-hyperconnected. 

  2. Every ∗-hyperconnected is ₢∗-hyperconnected. 

  3. Every ₢∗𝛼-hyperconnected is ₢∗-hyperconnected. 

  4. Every ₢∗𝛼𝑜-hyperconnected is ₢∗𝛼-hyperconnected. 

 

Proof.  

  1. Let Ⱥ is a ₢∗𝛼𝑜-hyperconnected, then 𝑐𝜄₢(Ⱥ) = Ꝡ then Ⱥ is a ∗-hyperconnected (since Ⱥ ∈

₢∗𝛼𝑜(Ꝡ). then  Ⱥ ∈ τ).    

  2. Let Ⱥ be a ∗-hyperconnected. This means that 𝑐𝜄₢(Ⱥ) = Ꝡ. So, Ꝡ − 𝑐𝜄₢(Ⱥ) = ∅ ∉ ₢. 

Therefore, Ⱥ is a ₢∗-hyperconnected.    

  3. Let Ⱥ be an open in ₢∗𝛼-hyperconnected that's mean that Ꝡ − 𝑐𝜄₢(Ⱥ) ∉ ₢
∗𝛼-hyperconnected  

(since every open set in ₢∗𝛼-hyperconnected is an open set in ₢∗-hyperconnected. So, Ꝡ −

𝑐𝜄₢(Ⱥ) ∉ ₢ .Therefore,  Ⱥ is a ₢
∗-hyperconnected. 

  4. Let Ⱥ be an open in ₢∗𝛼𝑜-hyperconnected. This means that 𝑐𝜄₢(Ⱥ) = Ꝡ. So, Ꝡ − 𝑐𝜄₢(Ⱥ) ∉

₢. Therefore, Ⱥ is a ₢∗𝑠-hyperconnected.    

 

 

The following diagram shows the relationship between the types of hyperconnected. 

 

 

 

                                                                  ∗-hyperconn𝑒cted 

 

 

                ₢∗𝛼𝑜-hyperconn𝑒cted                                                              ₢∗-hyperconn𝑒cted 

 

 

₢∗𝛼-hyperconnected                                                                      

                                                 

                                                 Diagram 1                          

hyperconnected space via grill space                                                        

                                                                      

 

 

 

. Conclusion4  

In this research we studied a connect space in the grill α -open topological spaces, we showed 

some examples and applied some theorems for this new sets. We also found some new 

properties of these sets. 

 

 

 

 



IHJPAS. 53 (4)2022 
 

219 

References     

 

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